Question

Show that $\left|\begin{array}{ccc}b+c& c+a& a+b\\ c+a& a+b& b+c\\ a+b& b+c& c+a\end{array}\right|=2\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$

Answer 1

$LHS=\left|\begin{array}{ccc}b+c& c+a& a+b\\ c+a& a+b& b+c\\ a+b& b+c& c+a\end{array}\right|$

$C_1\to C_1+C_2+C_3$

$=\left|\begin{array}{ccc}2(a+b+c)& c+a& a+b\\ 2(a+b+c)& a+b& b+c\\ 2(a+b+c)& b+c& c+a\end{array}\right|$

$=2\left|\begin{array}{ccc}a+b+c& c+a& a+b\\ a+b+c& a+b& b+c\\ a+b+c& b+c& c+a\end{array}\right|$

$C_2\to -C_2+C_1$

$C_3\to -C_3+C_1$

$=2\left|\begin{array}{ccc}a+b+c& b& c\\ a+b+c& c& a\\ a+b+c& a& b\end{array}\right|$

$C_1\to C_1-(C_2+C_3)$

$=2\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=RHS$